University - Complex numbers
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About this free course
This free course provides a sample of Level 3 study in Mathematics: www.open.ac.uk/courses/find/mathematics
This version of the content may include video, images and interactive content that may not be optimised for your device.
You can experience this free course as it was originally designed on OpenLearn, the home of free learning from The Open University: www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/complex-numbers/content-section-0.
There youll also be able to track your progress via your activity record, which you can use to demonstrate your learning.
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978-1-4730-1545-6 (.kdl)
978-1-4730-0777-2 (.epub)
You may have met complex numbers before, but not had experience in manipulating them. This course gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The course includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.
This OpenLearn course provides a sample of Level 3 study in Mathematics
After studying this course, you should be able to:
- perform basic algebraic manipulation with complex numbers
- understand the geometric interpretation of complex numbers
- know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.
You have almost certainly met complex numbers before, but you may well not have had much experience in manipulating them. In this course we provide you with an opportunity to gain confidence in working with complex numbers by working through a number of suitable problems.
Perhaps the most striking difference between real numbers and complex numbers is the fact that complex numbers have a two-dimensional character, arising from our definition of a complex number as an ordered pair of real numbers. This two-dimensional aspect of complex numbers leads to a most useful representation of them as points in the plane.
We know that the distance between points in a plane can be measured by the usual Euclidean measure of distance, and this leads us to the important modulus function for complex numbers. This function will clearly play an important role in complex analysis if the subject is to develop along lines resembling real analysis. Later we will see how the complex modulus can be used to generalise many of the limiting processes of real analysis.
Many proofs in real analysis use the concept of least upper bound. This concept depends in turn on the relation of inequality which is defined in terms of positive and negative numbers. We shall show in this course that it is not possible to define such a thing as a positive complex number and so least upper bound arguments will not carry over to complex analysis.
In real analysis, the idea of a least upper bound is used to develop the method of proof by repeated bisection, whose validity rests on the Nested Intervals Theorem. Because complex numbers are defined as pairs of real numbers, we are able to generalise this theorem to a Nested Rectangles Theorem, which will play a similar role in complex analysis to that of the Nested Intervals Theorem (or, equivalently, the least upper bound axiom) of real analysis.
There are three reading sections in this course, each of which includes a problems subsection. In various useful subsets of the complex number system are defined and the Nested Rectangles Theorem is proved.
In this section we shall define the complex number system as the set RR (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis.
In complex analysis we are concerned with functions whose domains and codomains are subsets of the set of complex numbers. As you probably know, this structure is obtained from the set RR by defining suitable operations of addition and multiplication. This reveals immediately one important difference between real analysis and complex analysis: in real analysis we are concerned with sets of real numbers, in complex analysis we are concerned with sets of ordered pairs
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